Each astronomical object is characterized by a
quantity called Mass-to-Light ratio, M/L. A convenient scaling of
M/L is done by using the value of the solar neighbourhood,
M /
L. Then,
observing M/L >
M
/ L
would imply the existence of DM of
unknown composition and origin. Note that in most cases the
evidence for M/L >
M /
L comes
from the gravitational effects
of the DM. Different classes of extragalactic objects (galaxies,
clusters, etc) are characterized by different M/L, indicating that
possibly a different fraction of the total mass of each type of object is
Dark.

Estimating the universal luminosity density (see further below) and using
the derived M/L values of each class of extragalactic object we can
estimate its contribution to the total
m. Furthermore, if the estimated
M/L is representative of the global universal value, then we can
derive the overall value of
m.

I will present below the basic ideas behind the determination of M/L
for the different extragalactic populations.

Spiral Galaxies: The Rotation curves of
spirals (see Fig.10) are
obtained by measuring Doppler-shifts of emission lines in HII regions, at
radio wavelengths using the 21-cm emission line of neutral Hydrogen or
using the CO-line and in the latter cases the rotation curve is measured
at several times the optical radius of a galaxy (for a recent review see
[162]).

One would expect the rotation curve to fall roughly as the surface
brightness and beyond a few length-scales to fall as
vrotr-1/2
(because most of the mass is rather centrally located) which is
not observed. The rotation curves are found to be flat as far as
they can be observed. From simple Newtonian Physics we have that:

and since vrot constant, we have
that M(r)
r, ie., mass
increases
linearly with distance beyond its optical radius, an indication for the
presence of dark matter. The average value of M/L found
for spiral galaxies out to ~ 20 kpc, is:

(75)

Figure 11. Rotation curves of 4 spiral
galaxies, one of which is the Galaxy (data taken from
[161]).

Note that the M/L value is an increasing function of outer
radius, implying the existence of an extended dark matter halo (cf.
[3]).

Elliptical Galaxies: In principle one can
invoke a similar
method with that of the spirals, if instead of the rotational velocity, the
stellar velocity dispersion is used. The amplitude of this velocity
measure is dictated by the gravitational potential of the elliptical
and thus the virial theorem can be used to determine its total
gravitating mass. However, stellar orbits are highly uncertain and the
velocity dispersion ellipsoid may not be isotropic (see discussion in
[16]).

Luckily, elliptical galaxies contain very hot gas (T ~ 5 ×
106K) which
emits X-rays. This gas constitutes about 10% of the observed mass in
stars. For a spherically symmetric galaxy in hydrostatic equilibrium
(v = 0) we have from Euler's equation that:

Using the ideal gas law (p =
kbT/m) with m the molecular mass and
kb the Boltzmann constant, we obtain that:

(76)

where T is the gas temperature, mp is the
proton mass and µ is the
mean molecular weight. Therefore if we measure the temperature and
density profiles we can find the total mass distribution
M(< r). Finally
the average mass-to-light ratio from this class of objects, out to ~ 20
kpc, is:

(77)

Groups of Galaxies: Groups of galaxies
containing a
few galaxies (3 - 10) are usually considered bound (due to the
high density of galaxies). Then, according to the Virial theorem
(2T + U = 0), a group with N galaxies having
velocity and position
vectors (relative to the centre of mass of the group)
vi and ri respectively, has:

However what we measure is not vi but only the
line-of-sight component
of the the velocity ui. Assuming isotropic orbits then
vi2
3<ui2>.
Assuming that the mass-to-light ratio of each member galaxy is the
same, mi / Li = M/L, we obtain:

(78)

where |Ri - Rj| is the projection of
|ri - rj| on the plane of the sky.

However this measure should be used only if N is large, or
otherwise statistically, ie, averaged over a large number of groups
to find the <M/L> of the ensemble. Note that this estimator is
very sensitive to the group-finding algorithm, selection procedure,
selection function corrections (ie., corrections to take into account the
fact that the galaxy density artificially decreases with depth in
magnitude/flux limited samples). Furthermore, projection effects (ie;
inclusion in the group of a foreground or background galaxy) is also
a source of error. For example the inclusion of a non-member
would result in an artificially higher M/L while conversely the
exclusion of a member (due to a high ui) would
underestimate the M/L.

Clusters of Galaxies: The most common
approach to measure
the cluster M/L is based on the assumption that clusters are
in virial equilibrium, for which we do have strong indications.

The first step in determining M/L is to measure the cluster velocity
dispersion accurately. Using the recent compilation of 395 R 0
ABELL/ACO cluster velocity dispersions
[167] I find:

However, this value is based even on clusters for which the velocity
dispersion was determined from a very small number of galaxy
redshifts, a fact which increases the possibility of assigning an
erroneous -value. This
can be seen in the left panel of Fig. 11 were we
plot the normalized
frequency distribution
for clusters with
Nz < 20 and Nz > 20. The
skewed distribution to large
-values for
Nz < 20 is evident, while the mean
-value is smaller than
for the Nz > 20 case. As a compromise between
having enough redshifts per cluster, in order to get a reliable
-value, and a large cluster
sample, we choose those with Nz > 20. For these 195
clusters I find that:

A consistent value of 940 ± 208 km/sec
has been found from the 16 high-redshift clusters of the CNOC project
[32]
and a variety of methods
used to define cluster membership and account for interlopers
[23].

Now from the Virial theorem, m =
22ra / G, we have:

(80)

where ra = 1.5 h-1 Mpc. Using a
number-galaxy weighted luminosity
estimation, and assuming an average cluster value in the optical of
L
1012h-2L (cf.
[4]),
we obtain the distribution of M/L values (see
right panel of Fig. 11). Since the distribution
is non-Gaussian we quote below the median and 68% confidence levels:

(81)

A more accurate method that can be used to estimate cluster mass is
based on the measurements of the X-ray emission from the ICM gas,
similar to the method used for individual elliptical galaxies (76).
Although not strictly correct, usually an isothermal cluster profile is used
(d ln T / d ln r = 0), which greatly simplifies
calculations. However, recent
experiments have shown that indeed the cluster Temperature does not
vary significantly with radius (d ln T / d ln r -0.7 ~ -0.8). Estimates of
M/L derived with this method are in general agreement with
(81).

Global luminosity density and m: We
estimate the value of
<L> using the galaxy luminosity function which is defined
such that
(L)dL is
the number density of galaxies having total luminosity in the
interval (L, L + dL). A good fit to the observed luminosity
function of
the field population of galaxies is provided by the Schechter
function
[153]:

In Table 2 we summarize the <M/L>
values found at different scales and the corresponding contribution to
m.

Since galaxy clusters are the deepest potential wells in the Universe
and they accumulate baryonic and DM from large volumes, it is expected
that their M/L ratio could represent the Universal value. This
view is
supported by the fact that the increasing trend of M/L with scale
(seen
in Table 2) reaches a plateau at the corresponding value of the clusters
(cf. [3],
[6]).
Therefore the universal value, as given by the clusters is: